#MathsInMinutes Day 34: Power Sets
#MathsInMinutes Day 34: Power Sets
Before we start with today´s entry I would like to heartfeltly thank @raymondsmith98, who has very generously donated 1 Tokens to support this ongoing series. These 1 Tokens will be used to boost the post in hopes of reaching a wider audience.
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The power set of a given set S is the set of all subsets of S, including S itself and the empty set. So if
S = {0,1}, then its power set, denoted P(S) is {Ø, {0}, {1}, {0,1}}.
The German mathematician Georg Cantor used power set to show that there are infinitely many different bigger and bigger classes of infinity, using an argument somewhat similar to, though predating, the barber paradox.
Cantor´s diagonal argument had already shown that there were at least two types of infinite set - countable, or listable, ones, and uncountable sets such as the continuum, the set of real numbers.
Cantor now showed that if S is an infinite set then its power set will always be bigger than S, in the sense that there is no way to map the elements of S to the elements of P(S) so that each element in one set is associated with one and only one element of the other set. In other words, the cardinality of P(S) is always larger than if S itself.
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